We provide a detailed study of the estimation of probability distributions---discrete and continuous---in a stringent setting in which data is kept private even from the statistician. We give sharp minimax rates of convergence for estimation in these locally private settings, exhibiting fundamental tradeoffs between privacy and convergence rate, as well as providing tools to allow movement along the privacy-statistical efficiency continuum. One of the consequences of our results is that Warner's classical work on randomized response is an optimal way to perform survey sampling while maintaining privacy of the respondents.

Branch-and-bound is a widely used method in combinatorial optimization, including mixed integer programming, structured prediction and MAP inference. While most work has been focused on developing problem-specific techniques, little is known about how to systematically design the node searching strategy on a branch-and-bound tree. We address the key challenge of learning an adaptive node searching order for any class of problem solvable by branch-and-bound. Our strategies are learned by imitation learning. We apply our algorithm to linear programming based branch-and-bound for solving mixed integer programs (MIP). We compare our method with one of the fastest open-source solvers, SCIP; and a very efficient commercial solver, Gurobi. We demonstrate that our approach achieves better solutions faster on four MIP libraries.

We consider online prediction problems where the loss between the prediction and the outcome is measured by the squared Euclidean distance and its generalization, the squared Mahalanobis distance. We derive the minimax solutions for the case where the prediction and action spaces are the simplex (this setup is sometimes called the Brier game) and the $\ell_2$ ball (this setup is related to Gaussian density estimation). We show that in both cases the value of each sub-game is a quadratic function of a simple statistic of the state, with coefficients that can be efficiently computed using an explicit recurrence relation. The resulting deterministic minimax strategy and randomized maximin strategy are linear functions of the statistic.

We consider the challenging practical problem of optimizing the power production of a complex of hydroelectric power plants, which involves control over three continuous action variables, uncertainty in the amount of water inflows and a variety of constraints that need to be satisfied. We propose a policy-search-based approach coupled with predictive modelling to address this problem. This approach has some key advantages compared to other alternatives, such as dynamic programming: the policy representation and search algorithm can conveniently incorporate domain knowledge; the resulting policies are easy to interpret, and the algorithm is naturally parallelizable. Our algorithm obtains a policy which outperforms the solution found by dynamic programming both quantitatively and qualitatively.

Biclustering is the analog of clustering on a bipartite graph. Existent methods infer biclusters through local search strategies that find one cluster at a time; a common technique is to update the row memberships based on the current column memberships, and vice versa. We propose a biclustering algorithm that maximizes a global objective function using message passing. Our objective function closely approximates a general likelihood function, separating a cluster size penalty term into row-and column-count penalties. Because we use a global optimization framework, our approach excels at resolving the overlaps between biclusters, which are important features of biclusters in practice. Moreover, Expectation-Maximization can be used to learn the model parameters if they are unknown. In simulations, we find that our method outperforms two of the best existing biclustering algorithms, ISA and LAS, when the planted clusters overlap. Applied to three gene expression datasets, our method finds coregulated gene clusters that have high quality in terms of cluster size and density.

The performance of flow matching and diffusion models can be greatly improved at inference time using reward alignment algorithms, yet efficiency remains a major limitation. While several algorithms were proposed, we demonstrate that a common bottleneck is the sampling method these algorithms rely on: many algorithms require to sample Markov transitions via SDE sampling, which is significantly less efficient and often less performant than ODE sampling. To remove this bottleneck, we introduce GLASS Flows, a new sampling paradigm that simulates a "flow matching model within a flow matching model" to sample Markov transitions. As we show in this work, this "inner" flow matching model can be retrieved from a pre-trained model without any re-training, combining the efficiency of ODEs with the stochastic evolution of SDEs. On large-scale text-to-image models, we show that GLASS Flows eliminate the trade-off between stochastic evolution and efficiency. Combined with Feynman-Kac Steering, GLASS Flows improve state-of-the-art performance in text-to-image generation, making it a simple, drop-in solution for inference-time scaling of flow and diffusion models.

Reinforcement learning with stochastic optimal control offers a promising framework for diffusion fine-tuning, where a pre-trained diffusion model is optimized to generate paths that lead to a reward-tilted distribution. While these approaches enable optimization without access to explicit samples from the optimal distribution, they require training on rollouts under the current fine-tuned model, making them susceptible to reinforcing sub-optimal trajectories that yield poor rewards. To overcome this challenge, we introduce TRee Search Guided TRajectory-Aware Fine-Tuning for Discrete Diffusion (TR2-D2), a novel framework that optimizes reward-guided discrete diffusion trajectories with tree search to construct replay buffers for trajectory-aware fine-tuning. These buffers are generated using Monte Carlo Tree Search (MCTS) and subsequently used to fine-tune a pre-trained discrete diffusion model under a stochastic optimal control objective. We validate our framework on single- and multi-objective fine-tuning of biological sequence diffusion models, highlighting the overall effectiveness of TR2-D2 for reliable reward-guided fine-tuning in discrete sequence generation.

The pretrain-transfer paradigm, which underpins the success of large language models (LLMs), has demonstrated the immense power of creating foundation models that learn generalizable representations from vast datasets. However, extending this paradigm to Operations Research (OR) problems on graph structures remains challenging due to the fundamental conflict between the statistical flexibility of language and the strict combinatorial constraints of graphs. To bridge this gap, we introduce the Graph Foundation Model (GFM), the first framework capable of solving all distance-based optimization problems on graph structures. By introducing the LLM-like self-supervised pre-training paradigm on the paths generated from random walks in the graph, GFM is compelled to internalize the graph's complex topological and combinatorial rules, where the connectivity of the structure itself can be treated as the supervisory signal. Unlike existing neural methods that learn complex and task-specific solving policies, our approach leverages the pre-trained GFM as a foundational model of the graph's intrinsic structure, which in turn enables a simple generative heuristic to tackle a diverse range of optimization challenges effectively. Comprehensive experiments on networks ranging from 20 to 893 nodes demonstrate that GFM achieves competitive performance against specialized solvers across a variety of distinct optimization task classes, while maintaining significantly faster inference times. Our work establishes a new paradigm of adapting the pretrain-transfer framework to graph optimization, opening the door for applying foundation model innovations to OR.

In classical AI, perception relies on learning state-based representations, while planning, which can be thought of as temporal reasoning over action sequences, is typically achieved through search. We study whether such reasoning can instead emerge from representations that capture both perceptual and temporal structure. We show that standard temporal contrastive learning, despite its popularity, often fails to capture temporal structure due to its reliance on spurious features. To address this, we introduce Combinatorial Representations for Temporal Reasoning (CRTR), a method that uses a negative sampling scheme to provably remove these spurious features and facilitate temporal reasoning. CRTR achieves strong results on domains with complex temporal structure, such as Sokoban and Rubik's Cube. In particular, for the Rubik's Cube, CRTR learns representations that generalize across all initial states and allow it to solve the puzzle using fewer search steps than BestFS, though with longer solutions. To our knowledge, this is the first method that efficiently solves arbitrary Cube states using only learned representations, without relying on an external search algorithm.

Integrating combinatorial optimization layers into neural networks has recently attracted significant research interest. However, many existing approaches lack theoretical guarantees or fail to perform adequately when relying on inexact solvers. This is a critical limitation, as many operations research problems are NP-hard, often necessitating the use of neighborhood-based local search heuristics. These heuristics iteratively generate and evaluate candidate solutions based on an acceptance rule. In this paper, we introduce a theoretically-principled approach for learning with such inexact combinatorial solvers. Inspired by the connection between simulated annealing and Metropolis-Hastings, we propose to transform problem-specific neighborhood systems used in local search heuristics into proposal distributions, implementing MCMC on the combinatorial space of feasible solutions. This allows us to construct differentiable combinatorial layers and associated loss functions. Replacing an exact solver by a local search strongly reduces the computational burden of learning on many applications. We demonstrate our approach on a large-scale dynamic vehicle routing problem with time windows.